2011/095) Prove: Let p and q be distinct primes, k be a natural number, and W be a natural number less than pq. Then W^(1+k(p-1)(q-1)) is congruent to W (mod pq

because p is a prime as per Formats Little theorem
now there are 3 cases and we deal as below
1) when w is coprime to p and q

w^(p-1) = 1 mod pwhen w is coprime to p

so w^(p-1)(q-1) = 1 mod p

similarly w^(p-1)(q-1) = 1 mod q

so w^(p-1)(q-1) = 1 mod pq

so w^k(p-1)(q-1) = 1 mod pq
or w^(1+k(p-1)(q-1)) = w mod pq

case 2
let w be a miultiple of p and not q

w^ t = w mod p as both are zero

so w ^(1+(k(p-1)(q-1)) = w mod p
as per argument in 1
w ^(1+(k(p-1)(q-1)) = w mod q

so w ^(1+(k(p-1)(q-1)) = w mod pq

3)
w is coprime to p and not q
same arguments as in 2

hence proved for all cases